Paraconsistent Logics and Translations

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we present some conservative translations involving classical logic, Lukasiewicz three-valued system L ₃, the intuitionistic system I ¹ and several paraconsistent logics, as for instance Sette's system P ¹, the D'Ottaviano and da Costa system J ₃ and da Costa's systems C _n, 1≤ n≤ω.     

A Hierarchy of Classical and Paraconsistent Logics

Journal of Philosophical Logic - In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a...     

Yes,Virginia, there Really are Paraconsistent Logics

B. H. Slater has argued that there cannot be any truly paraconsistent logics, because it's always more plausible to suppose whatever negation symbol is used in the language is not a real negation, than to accept the paraconsistent reading. In this paper I neither endorse nor dispute Slater's argument concerning negation; instead, my aim is to show that as an argument against paraconsistency, it misses (some of) the target. A important class of paraconsistent logics — the preservationist logics — are not subject to this objection. In addition I show that if we identify logics by means of consequence relations, at least one dialetheic logic can be reinterpreted in preservationist (non-dialetheic) terms. Thus the interest of paraconsistent consequence relations — even those that emerge from dialetheic approaches — does not depend on the tenability of dialetheism. Of course, if dialetheism is defensible, then paraconsistent logic will be required to cope with it. But the existence (and interest) of paraconsistent logics does not depend on a defense of dialetheism.     

Completeness and Cut-Elimination for First-Order Ideal Paraconsistent Four-Valued Logic

Studia Logica - In this study, we prove the completeness and cut-elimination theorems for a first-order extension F4CC of Arieli, Avron, and Zamansky’s ideal paraconsistent four-valued logic...     

Paraconsistent and Paracomplete Logics Based on k-Cyclic Modal Pseudocomplemented De Morgan Algebras

Studia Logica - The study of the theory of operators over modal pseudocomplemented De Morgan algebras was begun in papers [20] and [21]. In this paper, we introduce and study the class of modal...     

Paraconsistent algebras

The prepositional calculiC _n , 1 n introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum's algebra forC _n . C. Mortensen settled the problem, proving that no equivalence relation forC _n . determines a non-trivial quotient algebra.The concept of da Costa algebra, which reflects most of the logical properties ofC _n , as well as the concept of paraconsistent closure system, are introduced in this paper.We show that every da Costa algebra is isomorphic with a paraconsistent algebra of sets, and that the closure system of all filters of a da Costa algebra is paraconsistent.     

弗协调的弗雷格(英文)

本文表明,二阶弗协调概括与弗雷格的第五公理是足道的。也表明,如果等数关系是初始符号,那么通过弗协调推理可以从第五公理可以推出休谟原则。最后表明,弗协调的休谟原则不能用作逻辑主义数学的基础。     

Paraconsistent Logic

In some logics, anything whatsoever follows from a contradiction; call these logics explosive. Paraconsistent logics are logics that are not explosive. Paraconsistent logics have a long and fruitful history, and no doubt a long and fruitful future. To give some sense of the situation, I’ll spend Section 1 exploring exactly what it takes for a logic to be paraconsistent. It will emerge that there is considerable open texture to the idea. In Section 2, I’ll give some examples of techniques for developing paraconsistent logics. In Section 3, I’ll discuss what seem to me to be some promising applications of certain paraconsistent logics. In fact, however, I don’t think there’s all that much to the concept ‘paraconsistent’ itself; the collection of paraconsistent logics is far too heterogenous to be very productively dealt with under a single label. Perhaps that will emerge as we go.     

Paraconsistent Informational Logic

We introduce a Paraconsistent Informational Logic that formalizes the idea of conjectures which are acceptable as to the quality and the variety of the information that they convey with respect to a given theory T, even if they are classically inconsistent with T. The work constitutes an extension of a previously developed Informational Logic for classical frameworks, where a new notion of logical entropy measure H on formulas and on proofs plays a central role.     

On Paraconsistent Ethics

Abstract

Advances in modern logic allow us to admit that some ethical dilemmas are veridical contradictions. First, I offer two examples of such paradoxes and discuss their general form. Then the notion of paraconsistency is introduced to provide a coherent understanding of these as contradictions without lapsing into absurdity. With the groundwork in place, some features of an ethic embedded in paraconsistent logic are then shown to include a class of necessary and impossible acts called hypererogatory. Finally, some advantages of upholding such an ethic are discussed. This is a proposal to revaluate the form, rather than the content, of our normative judgements.     

A Paraconsistent Conditional Logic

Journal of Philosophical Logic - We develop a paraconsistent logic by introducing new models for conditionals with acceptive and rejective selection functions which are variants of Chellas’...     

Paraconsistent logic and model theory

The object of this paper is to show how one is able to construct a paraconsistent theory of models that reflects much of the classical one. In other words the aim is to demonstrate that there is a very smooth and natural transition from the model theory of classical logic to that of certain categories of paraconsistent logic. To this end we take an extension of da Costa'sC ₁ ⁼ (obtained by adding the axiom A A) and prove for it results which correspond to many major classical model theories, taken from Shoenfield [5]. In particular we prove counterparts of the theorems of o-Tarski and Chang-o-Suszko, Craig-Robinson and the Beth definability theorem.     

Real Analysis in Paraconsistent Logic

This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.     

Leibnizian Identity and Paraconsistent Logic

The standard Leibnizian view of identity allows for substitutivity of identicals and validates transitivity of identity within classical semantics. However, in a series of works, Graham Priest argues that Leibnizian identity invalidates both principles when formalized in paraconsistent semantics. This paper aims to show the Leibnizian view of identity validates substitutivity of identicals and transitivity of identity whether the logic is classical or paraconsistent. After presenting Priest's semantics of identity, I show what a semantic expression of Leibnizian identity does amount to. Then, I argue that Priest's semantic definition of identity is not Leibnizian. Finally, I offer a semantics characterization of identity in paraconsistent logic that is truly Leibnizian. I demonstrate that the correct formalization of Leibnizian identity in paraconsistent logic also validates substitutivity of identicals and transitivity of identity.     

A Paraconsistent Theory Of Belief Revision

This paper presents a theory of belief revision that allows people to come tobelieve in contradictions. The AGM theory of belief revision takes revision,in part, to be consistency maintenance. The present theory replacesconsistency with a weaker property called coherence. In addition to herbelief set, we take a set of statements that she rejects. These two sets arecoherent if they do not overlap. On this theory, belief revision maintains coherence.